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Mirrors > Home > MPE Home > Th. List > Mathboxes > submateqlem2 | Structured version Visualization version GIF version |
Description: Lemma for submateq 32447. (Contributed by Thierry Arnoux, 26-Aug-2020.) |
Ref | Expression |
---|---|
submateqlem2.n | ⊢ (𝜑 → 𝑁 ∈ ℕ) |
submateqlem2.k | ⊢ (𝜑 → 𝐾 ∈ (1...𝑁)) |
submateqlem2.m | ⊢ (𝜑 → 𝑀 ∈ (1...(𝑁 − 1))) |
submateqlem2.1 | ⊢ (𝜑 → 𝑀 < 𝐾) |
Ref | Expression |
---|---|
submateqlem2 | ⊢ (𝜑 → (𝑀 ∈ (1..^𝐾) ∧ 𝑀 ∈ ((1...𝑁) ∖ {𝐾}))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fz1ssnn 13478 | . . . . . 6 ⊢ (1...(𝑁 − 1)) ⊆ ℕ | |
2 | submateqlem2.m | . . . . . 6 ⊢ (𝜑 → 𝑀 ∈ (1...(𝑁 − 1))) | |
3 | 1, 2 | sselid 3943 | . . . . 5 ⊢ (𝜑 → 𝑀 ∈ ℕ) |
4 | 3 | nnge1d 12206 | . . . 4 ⊢ (𝜑 → 1 ≤ 𝑀) |
5 | submateqlem2.1 | . . . 4 ⊢ (𝜑 → 𝑀 < 𝐾) | |
6 | 4, 5 | jca 513 | . . 3 ⊢ (𝜑 → (1 ≤ 𝑀 ∧ 𝑀 < 𝐾)) |
7 | 2 | elfzelzd 13448 | . . . 4 ⊢ (𝜑 → 𝑀 ∈ ℤ) |
8 | 1zzd 12539 | . . . 4 ⊢ (𝜑 → 1 ∈ ℤ) | |
9 | submateqlem2.k | . . . . 5 ⊢ (𝜑 → 𝐾 ∈ (1...𝑁)) | |
10 | 9 | elfzelzd 13448 | . . . 4 ⊢ (𝜑 → 𝐾 ∈ ℤ) |
11 | elfzo 13580 | . . . 4 ⊢ ((𝑀 ∈ ℤ ∧ 1 ∈ ℤ ∧ 𝐾 ∈ ℤ) → (𝑀 ∈ (1..^𝐾) ↔ (1 ≤ 𝑀 ∧ 𝑀 < 𝐾))) | |
12 | 7, 8, 10, 11 | syl3anc 1372 | . . 3 ⊢ (𝜑 → (𝑀 ∈ (1..^𝐾) ↔ (1 ≤ 𝑀 ∧ 𝑀 < 𝐾))) |
13 | 6, 12 | mpbird 257 | . 2 ⊢ (𝜑 → 𝑀 ∈ (1..^𝐾)) |
14 | 2 | orcd 872 | . . . 4 ⊢ (𝜑 → (𝑀 ∈ (1...(𝑁 − 1)) ∨ 𝑀 = 𝑁)) |
15 | submateqlem2.n | . . . . . 6 ⊢ (𝜑 → 𝑁 ∈ ℕ) | |
16 | nnuz 12811 | . . . . . 6 ⊢ ℕ = (ℤ≥‘1) | |
17 | 15, 16 | eleqtrdi 2844 | . . . . 5 ⊢ (𝜑 → 𝑁 ∈ (ℤ≥‘1)) |
18 | fzm1 13527 | . . . . 5 ⊢ (𝑁 ∈ (ℤ≥‘1) → (𝑀 ∈ (1...𝑁) ↔ (𝑀 ∈ (1...(𝑁 − 1)) ∨ 𝑀 = 𝑁))) | |
19 | 17, 18 | syl 17 | . . . 4 ⊢ (𝜑 → (𝑀 ∈ (1...𝑁) ↔ (𝑀 ∈ (1...(𝑁 − 1)) ∨ 𝑀 = 𝑁))) |
20 | 14, 19 | mpbird 257 | . . 3 ⊢ (𝜑 → 𝑀 ∈ (1...𝑁)) |
21 | 3 | nnred 12173 | . . . . 5 ⊢ (𝜑 → 𝑀 ∈ ℝ) |
22 | 21, 5 | ltned 11296 | . . . 4 ⊢ (𝜑 → 𝑀 ≠ 𝐾) |
23 | nelsn 4627 | . . . 4 ⊢ (𝑀 ≠ 𝐾 → ¬ 𝑀 ∈ {𝐾}) | |
24 | 22, 23 | syl 17 | . . 3 ⊢ (𝜑 → ¬ 𝑀 ∈ {𝐾}) |
25 | 20, 24 | eldifd 3922 | . 2 ⊢ (𝜑 → 𝑀 ∈ ((1...𝑁) ∖ {𝐾})) |
26 | 13, 25 | jca 513 | 1 ⊢ (𝜑 → (𝑀 ∈ (1..^𝐾) ∧ 𝑀 ∈ ((1...𝑁) ∖ {𝐾}))) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 205 ∧ wa 397 ∨ wo 846 = wceq 1542 ∈ wcel 2107 ≠ wne 2940 ∖ cdif 3908 {csn 4587 class class class wbr 5106 ‘cfv 6497 (class class class)co 7358 1c1 11057 < clt 11194 ≤ cle 11195 − cmin 11390 ℕcn 12158 ℤcz 12504 ℤ≥cuz 12768 ...cfz 13430 ..^cfzo 13573 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-sep 5257 ax-nul 5264 ax-pow 5321 ax-pr 5385 ax-un 7673 ax-cnex 11112 ax-resscn 11113 ax-1cn 11114 ax-icn 11115 ax-addcl 11116 ax-addrcl 11117 ax-mulcl 11118 ax-mulrcl 11119 ax-mulcom 11120 ax-addass 11121 ax-mulass 11122 ax-distr 11123 ax-i2m1 11124 ax-1ne0 11125 ax-1rid 11126 ax-rnegex 11127 ax-rrecex 11128 ax-cnre 11129 ax-pre-lttri 11130 ax-pre-lttrn 11131 ax-pre-ltadd 11132 ax-pre-mulgt0 11133 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-reu 3353 df-rab 3407 df-v 3446 df-sbc 3741 df-csb 3857 df-dif 3914 df-un 3916 df-in 3918 df-ss 3928 df-pss 3930 df-nul 4284 df-if 4488 df-pw 4563 df-sn 4588 df-pr 4590 df-op 4594 df-uni 4867 df-iun 4957 df-br 5107 df-opab 5169 df-mpt 5190 df-tr 5224 df-id 5532 df-eprel 5538 df-po 5546 df-so 5547 df-fr 5589 df-we 5591 df-xp 5640 df-rel 5641 df-cnv 5642 df-co 5643 df-dm 5644 df-rn 5645 df-res 5646 df-ima 5647 df-pred 6254 df-ord 6321 df-on 6322 df-lim 6323 df-suc 6324 df-iota 6449 df-fun 6499 df-fn 6500 df-f 6501 df-f1 6502 df-fo 6503 df-f1o 6504 df-fv 6505 df-riota 7314 df-ov 7361 df-oprab 7362 df-mpo 7363 df-om 7804 df-1st 7922 df-2nd 7923 df-frecs 8213 df-wrecs 8244 df-recs 8318 df-rdg 8357 df-er 8651 df-en 8887 df-dom 8888 df-sdom 8889 df-pnf 11196 df-mnf 11197 df-xr 11198 df-ltxr 11199 df-le 11200 df-sub 11392 df-neg 11393 df-nn 12159 df-n0 12419 df-z 12505 df-uz 12769 df-fz 13431 df-fzo 13574 |
This theorem is referenced by: submateq 32447 |
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